Transcript ppt - Slides by Prof Christian

Outline – Stellar Evolution
I. Review Stellar Evolution (Quick)
II. Compact Objects & White Dwarfs
III. Lane-Emden Equation
IV. Chandrasekar Limit
V. Mixing Length Theory
A Census of the Stars
Faint, red dwarfs
(low mass) are
the most
common stars.
Bright, hot, blue
main-sequence
stars (highmass) are very
rare.
Giants and
supergiants
are extremely
rare.
Stellar Evolution
High mass star (> 8 M )
burn fuel faster, are brighter,
shorter life  SuperNova
Medium mass - (0.4 to 4 M) burn fuel moderatley, live long
 PN + WD
Low mass - (< 0.4 M)
burn fuel slowly - live long long
time!
Sun
Energy Transport Structure
Many Open Questions on stellar interior models
A-stars with X-rays; Models for low mass stars (BDs)
O & B-stars with Spots
8 Msun
Inner convective,
outer radiative
zone
4 Msun
Inner radiative,
outer convective
zone
CNO cycle dominant
PP chain dominant
Lane-Emden Equation
Use as a simple model of a star with polytropes in
Hydrostatic equilibrium
-or-
Think of it as Poisson’s Eq for a gravitational
potential of a Newtonian self gravitating, spherically
symmetric, polytropic fluid.
Solutions of Lane-Emden equation
for n=0 to 5
Mass Functions – The Eddington Solution
Red – standard solar model
Blue = n=3 polytrope
Black – linear
density law
Solar & n=3
Agree very well!
Solutions of Lane-Emden equation
n = 0, the density of the solution as a function of radius is constant,
ρ(r) = ρc. This is the solution for a constant density incompressible sphere.
n = 1 to 1.5 approximates a fully convective star, i.e. a very cool late-type
star such as a M, L, or T dwarf.
n = 3 is the Eddington Approximation.
There is no analytical solution for this value of n, but it is useful as it
corresponds to a fully radiative star, which is also a useful approximation
for the Sun.
n > 5, the binding energy is positive, and hence such a polytrope
cannot represent a real star.
Analytical solutions of Lane-Emden equation
for n=0,1,5
Expansion onto the Giant Branch
Expansion and
surface cooling during
the phase of an
inactive He core and
a H- burning shell
Sun will expand
beyond Earth’s orbit!
Red Giant Evolution
He-core gets smaller &
denser & hotter
4 H → He
He
GMm
r
Core - gravitational
contraction makes
more heat
H-burning shell
keeps dumping He
onto the core.
At shell, density and
gravity higher, must
burn faster to balance
More E, star expands
He fusion
through the
“Triple-Alpha
Process”
the next stage of nuclear
burning can begin in the core:
4He
+ 4He  8Be + g
8Be
+ 4He  12C + g
3 types of Compact Objects
White Dwarfs < 1.4 Msun (Chandrasekar limit)
Neutron Stars > 1.4 Msun < 3 Msun
Black Holes > 3 Msun
White Dwarfs - size of Earth (6000 km radius)
Neutron Stars - size of small city (10 km radius)
Black Holes - smaller* still than a city (< 10 km radius)!
(* really depends on mass)
Sizes of Stars & Stellar Remnants (1)
Sizes of Stars
Pauli’s Exclusion Principle:
No 2 electrons can have the same
Quantum mechanical state!
Sun
WD
BD
Sizes of Stars & Remnants
WD
White Dwarfs (2)
The more massive a white
dwarf, the smaller it is!
R ~ M-1/3
 Pressure becomes larger, until electron degeneracy
pressure can no longer hold up against gravity.
WDs with more than ~ 1.4 solar masses can not exist!
Chandrasekhar Limit = 1.4 Msun
White Dwarfs (3)
The more massive a white
dwarf, the smaller it is!
R ~ M-1/3
Summary of Stellar Evolution
Mixing Length Theory
The mixing length is conceptually analogous to the concept of mean free path in
thermodynamics: a fluid parcel will conserve its properties for a characteristic
length, , before mixing with the surrounding fluid.
Prandtl described that the mixing length:
“may be considered as the diameter of the masses of fluid moving as a whole
in each individual case; or again, as the distance traversed by a mass of this
type before it becomes blended in with neighbouring masses... “
Prandt 1925